$$ 5 . \quad $$ Determine the sum of all the factors of 100

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UpStudy AI · Accepted Answer

To determine the sum of all the factors of $$100$$, we can follow these steps:

1. **Find the prime factorization of $$100$$**:
   $$
   100 = 10 \times 10 = (2 \times 5) \times (2 \times 5) = 2^2 \times 5^2
   $$

2. **Use the formula for the sum of the factors**:
   If a number $$n$$ has the prime factorization $$p_1^{k_1} \times p_2^{k_2} \times \ldots \times p_m^{k_m}$$, the sum of the factors $$S(n)$$ can be calculated using the formula:
   $$
   S(n) = (1 + p_1 + p_1^2 + \ldots + p_1^{k_1})(1 + p_2 + p_2^2 + \ldots + p_2^{k_2}) \ldots (1 + p_m + p_m^2 + \ldots + p_m^{k_m})
   $$

3. **Apply the formula to $$100$$**:
   For $$100 = 2^2 \times 5^2$$:
   - For $$2^2$$:
     $$
     1 + 2 + 2^2 = 1 + 2 + 4 = 7
     $$
   - For $$5^2$$:
     $$
     1 + 5 + 5^2 = 1 + 5 + 25 = 31
     $$

4. **Calculate the sum of the factors**:
   Now, we multiply the results from the two prime factors:
   $$
   S(100) = 7 \times 31
   $$

Let's calculate $$7 \times 31$$.

$$
7 \times 31 = 217
$$

Thus, the sum of all the factors of $$100$$ is $$ \boxed{217} $$.
The sum of all the factors of 100 is 217.

\( 5 . \quad \) Determine the sum of all the factors of 100

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